Journal of Integer Sequences, Vol. 23 (2020), Article 20.9.6

Run Distribution Over Flattened Partitions


Olivia Nabawanda
Department of Mathematics
Makerere University
P. O. Box 7062
Kampala
Uganda

Fanja Rakotondrajao
Département de Mathématiques et Informatique
Université d'Antananarivo
Antananarivo
Madagascar

Alex Samuel Bamunoba
Department of Mathematics
Makerere University
P. O. Box 7062
Kampala
Uganda

Abstract:

The study of flattened partitions is an active area of current research. In this paper, our study unexpectedly leads us to the OEIS sequence A124324. We provide a new combinatorial interpretation of these numbers. A combinatorial bijection between flattened partitions over [n + 1] and the partitions of [n] is also given in a separate section. We introduce the numbers fn,k, which count the number of flattened partitions over [n] having k runs. We give recurrence relations defining them, as well as their exponential generating function in differential form. It should be appreciated if its closed form is established. We extend the results to flattened partitions where the first s integers belong to different runs. Combinatorial proofs are given.


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(Concerned with sequences A002182 A004394 A040047 A323256 A323257.)


Received February 10 2020; revised versions received July 1 2020; August 5 2020; October 15 2020. Published in Journal of Integer Sequences, October 15 2020.


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